PQ0003

prime_field_polynomial_negate_empty

Every pair of empty coefficient prefixes satisfies the operation, including modulus zero and arbitrary encodings.

Alpha v34 checked-use · first admitted v32 · independently kernel and Lean verified; not Stable

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

All coefficients retain the established highest-degree-first order. Trimming handles empty and all-zero prefixes; monic normalization requires a nonzero leading coefficient. Synthetic division has a nonempty input of length S n and a quotient of length n, unique in decoded values. Its coefficient recurrence, actual evaluation remainder and positive-degree drop are checked. General polynomial Euclidean division, gcd/Bezout, an arbitrary-convolution factor theorem, irreducible-polynomial existence and the full G091 prime-power-field endpoint remain open. These exact theorems are first admitted to Alpha v32; Stable remains unchanged.

Exact theorem in conservative defined notation

∀ p. ∀ ab. ∀ ac. ∀ rb. ∀ rc. FpCoefficientNegation(p,ab,ac,rb,rc,0)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

FpCoefficientNegation(p,ab,ac,rb,rc,L) · 1

Actual proof prerequisites

lt_not_le · checked external prerequisitezero_le · checked external prerequisite
Original expanded first-order statement 
forall p ab ac rb rc. forall pfs_index_negate_empty. (exists pfa_gap_negate_emptyindex. pfa_gap_negate_emptyindex + S (pfs_index_negate_empty) = (0)) -> exists pfs_source_negate_empty pfs_result_negate_empty. ((((exists ff_h_pfp_negate_emptysource. ff_h_pfp_negate_emptysource + S (pfs_source_negate_empty) = S ((S (pfs_index_negate_empty)) * ac)) /\ exists ff_q_pfp_negate_emptysource. ab = ff_q_pfp_negate_emptysource * S ((S (pfs_index_negate_empty)) * ac) + (pfs_source_negate_empty))) /\ (((((exists ff_h_pfp_negate_emptyresult. ff_h_pfp_negate_emptyresult + S (pfs_result_negate_empty) = S ((S (pfs_index_negate_empty)) * rc)) /\ exists ff_q_pfp_negate_emptyresult. rb = ff_q_pfp_negate_emptyresult * S ((S (pfs_index_negate_empty)) * rc) + (pfs_result_negate_empty))) /\ ((((exists pfa_gap_negate_emptyoperationadditionleft. pfa_gap_negate_emptyoperationadditionleft + S (pfs_source_negate_empty) = (p)) /\ (((exists pfa_gap_negate_emptyoperationadditionright. pfa_gap_negate_emptyoperationadditionright + S (pfs_result_negate_empty) = (p)) /\ ((((exists pfa_gap_negate_emptyoperationadditionresultbound. pfa_gap_negate_emptyoperationadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_emptyoperationadditionresultcongruence pfa_offset_right_negate_emptyoperationadditionresultcongruence. ((pfs_source_negate_empty) + (pfs_result_negate_empty)) + (p) * pfa_offset_left_negate_emptyoperationadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_emptyoperationadditionresultcongruence)))))))))))))

Complete tactic proof in conservative notation

All 14 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

14 script commands · 3 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–7

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro p
  2. L2
    intro ab
  3. L3
    intro ac
  4. L4
    intro rb
  5. L5
    intro rc
  6. L6
    intro i
  7. L7
    intro hi
02Separate the logical casesL8–8

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L8
    exfalso
03Use earlier factsL9–14

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L9
    specialize lt_not_le (i)
  2. L10
    specialize lt_not_le (0)
  3. L11
    apply lt_not_le
  4. L12
    exact hi
  5. L13
    specialize zero_le (i)
  6. L14
    apply zero_le

Library-wide reading audit

Original defined command ledger · 14 lines
  1. 0001intro p
  2. 0002intro ab
  3. 0003intro ac
  4. 0004intro rb
  5. 0005intro rc
  6. 0006intro i
  7. 0007intro hi
  8. 0008exfalso
  9. 0009specialize lt_not_le (i)
  10. 0010specialize lt_not_le (0)
  11. 0011apply lt_not_le
  12. 0012exact hi
  13. 0013specialize zero_le (i)
  14. 0014apply zero_le